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(d)   If a, b e A are self-adjoint, show that p(a2) < p(a2 -f b2). (Remark that
p(a2 + b2)e - a2 = (p(a2 + b2)e -a2- b2) + b2,

and use (c).) Deduce that p(x* + x) <; 2/>(x) for all * e A.

(e)    Deduce from (b) and (d) that p(x + y) ^p(x) +p(y) for all x, y E A. If A is
without radical, it follows that p defines on A a normed algebra structure, for which
the topology is coarser than that defined by the norm \\x\\. These two topologies
coincide only if A is complete relative to the norm p.

(f)    Show that if A is without radical, the function x*-+x* is continuous on A. (Use
the closed graph theorem, by remarking that if a sequence (xn) is such that xn - 0
and x% -+y, then p(x*) and p(y  x*) both tend to 0; then use (e).)

(g)    Show that p is a continuous function on A. (Use (f), by remarking that there
exists a constant c>0 such that ||TT(;C*)|| ^c|br(jc)||, and that pA(x) = pA/ttW*)).)
(h)   Show that, for each x e A, SpOt**) is contained in [0, + oo[. (Argue by contra-
diction: suppose that there exists x e A such that  1 e Sp(x*x). Write z  x*x, and
for each positive integer n let wn be a self-adjoint element of A which commutes with z,

and is such that w% = z2 + - e and Sp(wn) <= [0, -j- oo [ (Problem 17). Let bn = wn  z.

By considering a commutative Banach subalgebra B of A containing wn and z, and
such that the spectra of z, wn , and bn are the same in B and in A, show that

Show also that p(bn) <* 1 + 2p(z) = a, independent of n (use (c)). Put yn  xbn , so
that y*yn =  b2nwn + Wn)bn; deduce that Sp(y$yn) is contained in the interval
] oo, a//i], and then that Sp(jw y%) is contained in the interval [ a///, + oo[ (use (c));
hence that p(y%yn) ^ ot/n (Section 1 5.2, Problem 2(b)), and consequently that
IxMjO! ^oc/n for all characters x f B- Finally, obtain a contradiction by noting
that ylyn = b2z, that x(wn) ^ 0 and that there exists a character x such that x(z) =  1 .)

19. Let A be a Banach algebra with unit element, and let xt~+x* be a (not necessarily
continuous) involution on A. Show that the following properties of A are equivalent :

(a)   A is hermitian.

(j8)   p(x) ^p(x) for all normal x e A.

(y)   p(x)  p(x) for all normal x e A.

(S)   p(** + *)g2X*)forallx6A.

(e)   p(x + y) <p(x) + Xy) for all x, y e A.

() The set of numbers p(x), where * runs through the unitary elements of A,
is bounded.

(r))   For each x e A the element e + x*x is invertible in A.

(To show that (17) implies (a), argue by contradiction by showing that the
spectrum of a hermitian element cannot contain the number /. To show that (y)
implies (a), argue as in (15.4.12). To show that (8) implies (/3), remark that (8)
implies that p(x)n = p(xn) g 2p(x)n for x normal and n ;> 1 . Finally, to show that ()
implies (a), notice first that () implies that Sp(;c) c U for all unitary elements x, by
considering the powers xn (n e Z). Next observe that if a is self-adjoint and p(a) < I ,
there exists a self-adjoint element b, commuting with a, such that b2 = <?  a2
(Problem 17), hence a + & is unitary. Then consider a commutative Banach subalgebra
containing a and 6 and such that the spectra of a, b and a + ib are the same in A and B).ments cz, b of A,